continuity AWS
CONTINUITY Instructor: Mohammad Mashayekhi
Outline: ■ What is continuity? ■ Different types of discontinuity ■ Basic rules of continuity ■ Examples
What is continuity? § A function 𝑓(𝑥) is said to be continuous at 𝑥 = 𝑎 if: 1) lim. 𝑓(𝑥) = lim/ 𝑓(𝑥) = 𝐿 +→-
+→-
(𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑓(𝑥) 𝑎𝑡 𝑥 = 𝑎 𝑒𝑥𝑖𝑠𝑡𝑠)
2) 𝑓(𝑥) 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑡 𝑥 = 𝑎 3) lim 𝑓(𝑥) = 𝑓(𝑎) +→-
§ A function 𝑓(𝑥) is said to be continuous over an interval (𝐴, 𝐵) if it is continuous for every point in this interval. § A function 𝑓(𝑥) is said to be continuous if it is continuous for all points in ℝ.
What is continuity? § Some popular continuous functions:
ü Polynomials are continuous functions over ℝ. ü Trigonometric functions are continuous over their domain. ü A rational function is continuous over ℝ excepts points which makes its denominator equal to zero (over its domain). ü Root and powers are continuous over their domain. ü Exponential and logarithmic functions are continuous over their domain.
Quiz! ■ Which of the following statements is correct about 𝑓(𝑥): (𝑥 + 1)(𝑥 − 2)D 𝑓 𝑥 = (𝑥 − 2)(3 − 𝑥)
a) 𝑓 𝑥 is continuous over ℝ b) 𝑓 𝑥 is discontinuous at 𝑥 = −1 c) 𝑓 𝑥 is discontinuous at 𝑥 = 2 d) 𝑓 𝑥 is continuous at points which make numerator equal to zero.
Quiz solution: ■
Which of the following statements is correct about 𝑓(𝑥): (𝑥 + 1)(𝑥 − 2)D 𝑓 𝑥 = (𝑥 − 2)(3 − 𝑥)
Correct answer: c) 𝑓 𝑥 is discontinuous at 𝑥 = 2
a) 𝑓 𝑥 is continuous over ℝ b) 𝑓 𝑥 is discontinuous at 𝑥 = −1 c) 𝑓 𝑥 is discontinuous at 𝑥 = 2 d) 𝑓 𝑥 is continuous at points which make numerator equal to zero.
Outline: ■ What is continuity? ■ Different types of discontinuity ■ Basic rules of continuity ■ Examples
Different types of discontinuity: a) Infinite limits
b) Jump
c) Hole
Basic rules of continuity: ■ Basic rules that could be used to find the continuity of more complex functions: If 𝑓 𝑥 and g 𝑥 are continuous functions at 𝑥 = 𝑎 and 𝑐 is a constant: ü 𝑓 𝑥 ± 𝑔 𝑥 is also continuous at 𝑥 = 𝑎 ü 𝑓 𝑥 . 𝑔 𝑥 is also continuous at 𝑥 = 𝑎 ü
L(+) M(+)
is also continuous at 𝑥 = 𝑎 except if g 𝑎 = 0
ü 𝑐. 𝑓 𝑥 is also continuous at 𝑥 = 𝑎 Example: Find all 𝑥 values for which 𝑓 𝑥 =
DO P QRST + +U VW
continuous.
Basic rules of continuity: ■ One more rule: 𝑓 𝑔(𝑥) is continuous at 𝑥 = 𝑎 if: 1) g 𝑥 is continuous at 𝑥 = 𝑎
2) 𝑓 𝑥 is continuous at 𝑥 = 𝑔(𝑎)
Example: Find all the points for which 𝑓 𝑥 =
ln (𝑥) is continuous.
Outline: ■ What is continuity? ■ Different types of discontinuity ■ Basic rules of continuity ■ Examples
Example 1: § Find 𝑎 such that 𝑓(𝑥) is continuous at 𝑥 = 1 𝑎𝑥 D + 2 𝑥 ≥ 1 𝑓 𝑥 =Z ] −2 sin 𝑥 < 1
Ø Solution:
+QW
Example 2: § Find all 𝑥 values for which 𝑓 𝑥 = Ø Solution:
W is V_`R +
continuous.
Cos x
Example 3: § Find 𝑎 and 𝑏 such that the following function is continuous at 𝑥 = 0:
Ø Solution:
𝑎D sin 𝑥 + 𝑎𝑒 + + 𝑏 𝑥 > 0 𝑔 𝑥 = b 3 𝑥 = 0 d+QD 𝑥 < 0 e+Ve
𝑎D sin 𝑥 + 𝑎𝑒 + + 𝑏 𝑥 > 0 3 𝑥 = 0 𝑔 𝑥 = 5𝑥 + 2𝑎 𝑥 < 0 3𝑥 − 3
THANKS FOR YOUR TIME!
Outline: ■ What is continuity? ■ Different types of discontinuity ■ Basic rules of continuity ■ Examples
What is continuity? § A function 𝑓(𝑥) is said to be continuous at 𝑥 = 𝑎 if: 1) lim. 𝑓(𝑥) = lim/ 𝑓(𝑥) = 𝐿 +→-
+→-
(𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑓(𝑥) 𝑎𝑡 𝑥 = 𝑎 𝑒𝑥𝑖𝑠𝑡𝑠)
2) 𝑓(𝑥) 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑡 𝑥 = 𝑎 3) lim 𝑓(𝑥) = 𝑓(𝑎) +→-
§ A function 𝑓(𝑥) is said to be continuous over an interval (𝐴, 𝐵) if it is continuous for every point in this interval. § A function 𝑓(𝑥) is said to be continuous if it is continuous for all points in ℝ.
What is continuity? § Some popular continuous functions:
ü Polynomials are continuous functions over ℝ. ü Trigonometric functions are continuous over their domain. ü A rational function is continuous over ℝ excepts points which makes its denominator equal to zero (over its domain). ü Root and powers are continuous over their domain. ü Exponential and logarithmic functions are continuous over their domain.
Quiz! ■ Which of the following statements is correct about 𝑓(𝑥): (𝑥 + 1)(𝑥 − 2)D 𝑓 𝑥 = (𝑥 − 2)(3 − 𝑥)
a) 𝑓 𝑥 is continuous over ℝ b) 𝑓 𝑥 is discontinuous at 𝑥 = −1 c) 𝑓 𝑥 is discontinuous at 𝑥 = 2 d) 𝑓 𝑥 is continuous at points which make numerator equal to zero.
Quiz solution: ■
Which of the following statements is correct about 𝑓(𝑥): (𝑥 + 1)(𝑥 − 2)D 𝑓 𝑥 = (𝑥 − 2)(3 − 𝑥)
Correct answer: c) 𝑓 𝑥 is discontinuous at 𝑥 = 2
a) 𝑓 𝑥 is continuous over ℝ b) 𝑓 𝑥 is discontinuous at 𝑥 = −1 c) 𝑓 𝑥 is discontinuous at 𝑥 = 2 d) 𝑓 𝑥 is continuous at points which make numerator equal to zero.
Outline: ■ What is continuity? ■ Different types of discontinuity ■ Basic rules of continuity ■ Examples
Different types of discontinuity: a) Infinite limits
b) Jump
c) Hole
Basic rules of continuity: ■ Basic rules that could be used to find the continuity of more complex functions: If 𝑓 𝑥 and g 𝑥 are continuous functions at 𝑥 = 𝑎 and 𝑐 is a constant: ü 𝑓 𝑥 ± 𝑔 𝑥 is also continuous at 𝑥 = 𝑎 ü 𝑓 𝑥 . 𝑔 𝑥 is also continuous at 𝑥 = 𝑎 ü
L(+) M(+)
is also continuous at 𝑥 = 𝑎 except if g 𝑎 = 0
ü 𝑐. 𝑓 𝑥 is also continuous at 𝑥 = 𝑎 Example: Find all 𝑥 values for which 𝑓 𝑥 =
DO P QRST + +U VW
continuous.
Basic rules of continuity: ■ One more rule: 𝑓 𝑔(𝑥) is continuous at 𝑥 = 𝑎 if: 1) g 𝑥 is continuous at 𝑥 = 𝑎
2) 𝑓 𝑥 is continuous at 𝑥 = 𝑔(𝑎)
Example: Find all the points for which 𝑓 𝑥 =
ln (𝑥) is continuous.
Outline: ■ What is continuity? ■ Different types of discontinuity ■ Basic rules of continuity ■ Examples
Example 1: § Find 𝑎 such that 𝑓(𝑥) is continuous at 𝑥 = 1 𝑎𝑥 D + 2 𝑥 ≥ 1 𝑓 𝑥 =Z ] −2 sin 𝑥 < 1
Ø Solution:
+QW
Example 2: § Find all 𝑥 values for which 𝑓 𝑥 = Ø Solution:
W is V_`R +
continuous.
Cos x
Example 3: § Find 𝑎 and 𝑏 such that the following function is continuous at 𝑥 = 0:
Ø Solution:
𝑎D sin 𝑥 + 𝑎𝑒 + + 𝑏 𝑥 > 0 𝑔 𝑥 = b 3 𝑥 = 0 d+QD 𝑥 < 0 e+Ve
𝑎D sin 𝑥 + 𝑎𝑒 + + 𝑏 𝑥 > 0 3 𝑥 = 0 𝑔 𝑥 = 5𝑥 + 2𝑎 𝑥 < 0 3𝑥 − 3
THANKS FOR YOUR TIME!
